Test- a 33.pdf - f(b − f(a = f′(c(b − a Proof...

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f(b) f(a) = f (c)(b a). Proof. Consider F : [a, b] R defined by F(x) = f(x) f(a) s(x a), where s := f(b) f(a) . b a Then F(a) = 0 and our choice of the constant s is such that F(b) = 0. So the Rolle Theorem applies to F , and as a result, there is c (a, b) such that F (c) =
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0. This implies that f (c) = s, as desired. ⊓⊔ Remark 4.21. If we write b = a + h, then the conclusion of the MVT may be stated as follows: f(a+h)=f(a)+hf (a+ θ h) forsome θ (0,1). The equivalence with the MVT is easily
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